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I've been taught that things that have mass "create" a gravitational pull, and that an electron has mass, even though it's really small. So I was thinking, does the electron's gravitational pull increase a circuit's gravitational pull (again, even if by a small amount)?

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Regardless of whether or not the electrons had a significant effect on the gravitational pull, turning a circuit on does not produce electrons out of nothing. The electrons that are present in the circuit were already there before the circuit was turned on. Turning the circuit on just causes the electrons to start moving around the circuit.

So no, turning a circuit on does not change its gravitational pull, at least classically. I am sure one could make some relativistic energy argument about how changes in energy could technically change some gravitational effects, but this would be negligible and beyond the scope of the question probably.

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Your hand will cause some gravitational pull. No new electrons will appear in the system.

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